Question: Evaluate the following expression. Your answer must be exact. $\left(\dfrac{3}{2}+\dfrac{3\sqrt{3}i}{2}\right)^3=$
The Strategy The easiest way to find $z^{n}$ for a complex number $z=({a}+{b}i)$ is using its modulus and argument. Therefore, our solution will consist of the following steps: Find the modulus and argument of $z$. [How is this done, in general?] Find the modulus and argument of $z^{n}$. [How is this done, in general?] Find the rectangular form $z^{n}$. Find the modulus and argument of $\left(\dfrac{3}{2}+\dfrac{3\sqrt{3}i}{2}\right)$ $\left({\dfrac{3}{2}}+{\dfrac{3\sqrt{3}}{2}}i\right)$ is of the form $({a}+{b}i)$, where ${a=\dfrac{3}{2}}$ and ${b=\dfrac{3\sqrt{3}}{2}}$. Therefore: $\begin{aligned}r&=\sqrt{{a}^2 + {b}^2} \\\\&=\sqrt{ \left({\dfrac{3}{2}}\right)^2 + \left({\dfrac{3\sqrt{3}}{2}}\right)^2} \\\\&=\sqrt{{\dfrac{9}{4}}+{\dfrac{27}{4}}} \\\\&=3\end{aligned}$ Using the arctangent formula, we have: $\begin{aligned}\theta&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{\dfrac{3\sqrt{3}}{2}}}{{{\dfrac{3}{2}}}}\right) \\\\&=60^\circ\end{aligned}$ Since both ${a=\dfrac{3}{2}}$ and ${b=\dfrac{3\sqrt{3}}{2}}$ are positive, $\left(\dfrac{3}{2}+\dfrac{3\sqrt{3}i}{2}\right)$ lies in Quadrant $1$. Therefore, $\theta$ must be between $0^\circ$ and $90^\circ$, so our answer matches our requirements. Find the modulus and argument of $\left(\dfrac{3}{2}+\dfrac{3\sqrt{3}i}{2}\right)^3$ We found that the modulus and argument of $\left({\dfrac{3}{2}}+{\dfrac{3\sqrt{3}}{2}}i\right)$ are $3$ and $60^\circ$. Therefore, the modulus and argument of $\left({\dfrac{3}{2}}+{\dfrac{3\sqrt{3}}{2}}i\right)^3$ are $3^3=27$ and $(60^\circ)\cdot3=180^\circ$. Find the rectangular form of $\left(\dfrac{3}{2}+\dfrac{3\sqrt{3}i}{2}\right)^3$ Since the argument is $180°$, we know the number lies on the negative side of the real number axis and is therefore a negative real number. Since the modulus is $27$, our solution is $-27$. [What does this look like graphically?] [How do we find this algebraically?] Summary $\left(\dfrac{3}{2}+\dfrac{3\sqrt{3}i}{2}\right)^3=-27$